Step by step
The question
Question: solve |2x + 1| = |x + 5|.
Both bars give how far from 0.
Two numbers the same distance from 0 are equal, or opposite.
Like 3 and minus 3.
Careful!
The minus is for the whole of x + 5.
Every sign in the bracket changes.
1 Two cases
So 2x + 1 = x + 5, or 2x + 1 = −(x + 5).
We keep the bracket: the minus is for all of it.
2 Solve the first case
First case: 2x + 1 = x + 5.
3 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the x from the right side.
To remove x, we take away x.
We do it on both sides, so they stay equal.
x and − x make zero.
Cross them out.
2x − x = x.
Now we have x + 1 = 5.
4 Remove the + 1
We want x by itself.
The + 1 is in the way.
To remove + 1, we take away 1.
We do it on both sides, so they stay equal.
+ 1 and − 1 make zero.
Cross them out.
5 − 1 = 4.
So x = 4!
5 Solve the second case
Second case: 2x + 1 = −(x + 5).
6 Open the brackets
A minus in front of a bracket changes the sign of every part inside.
x becomes −x.
5 becomes −5.
7 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the −x from the right side.
To remove −x, we add x.
We do it on both sides, so they stay equal.
− x and + x make zero.
Cross them out.
2x + x = 3x.
Now we have 3x + 1 = −5.
We want x by itself.
Two things are stuck to x: times 3 and + 1.
The + 1 is on the outside, so it comes off first.
Then the times 3.
8 Remove the + 1
To remove + 1, we take away 1.
We do it on both sides, so they stay equal.
+ 1 and − 1 make zero.
Cross them out.
−5 − 1 = −6.
Now we have 3x = −6.
9 Remove the 3
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just x is left.
−6 divided by 3 is −2.
So x = −2!
10 Write the answers
So x = 4 or x = −2.
11 Check
Let's check: put x = 4 in.
The left bars give 9.
The right bars give 9 too.
The same!
Now put x = −2 in.
The left bars give 3.
The right bars give 3 too.
The same!
Both answers work!
Let's look back.
First: two cases.
Then: solve the first case.
Then: get x on one side.
Then: remove the + 1.
Then: solve the second case.
Then: open the brackets.
Then: get x on one side.
Then: remove the + 1.
Then: remove the 3.
Then: write the answers.
Last: check.